Algebraic bivariant K-theory and nuclear modules

We will introduce the purely algebraic version of Kasparov's KK-theory via the category of localizing motives. Namely, this category is the target of the universal localizing invariant of small stable infinity-categories (over some base ring spectrum), commuting with filtered colimits. Now, the KK-theory spectrum for a pair of categories $A$ and $B$ is just the spectrum of morphisms from the motive of $A$ to the motive of $B$. We will explain how to compute this KK-theory, in particular the K-homology. As a special case we will recover the comparison result between the two approaches to K-theory of formal schemes: the classical continuous K-theory is equivalent to the K-theory of the category of nuclear modules. Another closely related special case is the identification of the K-homology of a smooth scheme with its "K-theory with proper supports".